An accelerated life test model with the inverse power law

Reliability Engineering and System Safety 24 (1989) 223-230

An Accelerated Life Test Model with the Inverse Power Law D. S. Bai & S. W. Chung Department of Industrial Engineering, Korea Advanced Institute of Science and Technology, PO Box 150, Chongryang, Seoul, Republic of Korea

(Received 19 May 1988; accepted 9 June 1988)

A BSTRA CT Constant and progressive stress accelerated life test models based on the inverse power law and Weibull lifetime distributions are proposed m which the Weibull scale parameter at use condition has a gamma prior distribution and other parameters are unknown constants. Bayes and maximum likelihood methods are used to estimate the model parameters. Monte Carlo study is performed to investigate the behaviour of the estimators.

NOTATION g(x;a,b)

G a m m a p d f with p a r a m e t e r s a a n d b; g ( x ; a , b ) = ba/F(a)x ~- l exp ( - bx) k N u m b e r o f c o n s t a n t stress levels m N u m b e r o f items tested u n d e r progressive stress n T o t a l n u m b e r o f items tested u n d e r multiple c o n s t a n t stresses, i.e. n = y k= 1 r/i ni N u m b e r o f items tested at V~, i = 1,..., k t~~' 27~1 t ~ , i = 1. . . . . k tit Failure time o f t h e j t h item at stress level V,. u n d e r multiple c o n s t a n t stresses, i = l ..... k; j = 1..... n i Vi Stress test level u n d e r multiple c o n s t a n t stresses, i = 1..... k 1/", Stress level at use c o n d i t i o n W(t;2,fl) Weibull c d f with p a r a m e t e r s 2 and fl; W(t;;~,fl)= 1 - exp ( - ~.t~) 223 Reliabilio. Engineering and System Safi, ty 0951-8320/89/$03'50 ~, 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain

I). S. Bai. S. W. ('hun~

224

Failure time of the jth item subject to progressive stress,

.Vi

j -= 1 . . . . .

m

Rate of stress increase with time under progressive stress

P Assumptions 11) (2) (3) (4)

At constant stress level V~, life time follows a Weibull distribution with parameters 2~ and [L The Weibull scale parameter ), at use condition has a gamma prior distribution with parameters a and h. The scale parameter of the Weibull distribution at stress level V~is a power function of the stress ).(Vi/V,) p. The shape parameter [ / o f the Weibull distribution and the power parameter p of the power function are unknown constants.

! INTRODUCTION Accelerated life testing of a product or material is used to obtain information quickly on its life distribution. Test units are run under severe conditions and fail sooner than under use condition. The accelerated failure times are then extrapolated to estimate the life distribution under use condition. It may be quicker and cheaper than testing at use condition, which is usually impractical because of long product life. In life testing of a product, it is c o m m o n to have supplemental information on the product that is tested such as engineering designs, engineering judgements and operating experiences with similar products. Bayesian accelerated life test models which combine supplemental information on the product with its accelerated life test data have been studied by several authors. Proschan and Singpurwalla, 1'2 and Shaked and Singpurwalla 3 proposed Bayesian accelerated life test models with nonparametric assumptions about the life distribution at each stress level and/or the relationship between stress level and product life. D e G r o o t and Goel 4 presented a partially accelerated life test model in which a unit is first tested at use condition and, if it does not fail for a specified time, it is put under one accelerated condition. They considered a model in which the life distributions at both use and accelerated conditions are exponential and the acceleration factor, which is defined as the ratio of the hazard rate at accelerated condition to the hazard rate at use condition, has a gamma distribution with scale parameter proportional to use condition hazard rate whose priror distribution is also gamma. Pathak and Zimmer 5 considered a model in which units are tested until failure at only one accelerated condition. The model also assumes that the life distributions at both use and

An accelerated life test model with the inverse power law

225

accelerated conditions are exponential, and the hazard rate at use condition has a gamma prior distribution. In this paper, we consider an accelerated life test model in which units are tested at multiple constant stress or progressive stress conditions and the inverse power law is obeyed. The life distributions at both use condition and accelerated conditions are Weibull distribution, and its scale parameter at use condition has a gamma prior distribution. Bayes and maximum likelihood methods are used to estimate the parameters.

2 THE MODEL

2.1 Life test under multiple constant stresses M a x i m u m likelihood estimators of p and fl are obtained when life test is conducted under multiple constant stress conditions. To obtain an estimator of 2, they are then substituted for p and fl in the expression for the posterior mean of ). given the test data.

2.1.1 Test procedure The n; units are put on test at V~ level of stress, i = 1..... k, and the stress is applied constantly at each level until all units fail. 2.1.2 Parameter estimation From the conditional distribution of failure times given 2 and the prior distribution of 2, the likelihood function L(p,//; tl i,..., tk,k) is given by L(p, f l ; t l l , t l 2

.....

tknk) =

L(P, fl, 2;tll,tl2,...,tk,k)d), k

ni

i-lj=l

x 2" - l exp ( - b2)ba/F(a) d2 k

i~l

k

nt

i=lj=l k

i=1

226

D. S. Bai, S. W. Chung

The posterior distribution of 2 is obtained as the ratio of L ( p , [ L ) . ; t ~ ..... tk.~) to L(p, f l ; t ~ ..... tk, ~) and turns out to be a gamma distribution with parameters, {b + y~: ~ ( V~./ Vu) Pt "~! i ,.and ( n + a ) . The posterior mean of 2 is given by k

Ep.~[,;.;t,, . . . . . t k , J = ( n + a ) /

f X h+

(V~/V~)Ptl t~

i=l

/

(2)

which is the Bayes estimator of,;, under squared error loss when p and/3 are known constants. The m a x i m u m likelihood estimator (/~,/~) of (p, fl) is obtained by solving the two partial derivatives of the log L(p, fl; t~ ~. . . . . tk,~) with respect to p and [:I simultaneously. E~,~[tll . . . . . tk.~] obtained by substituting (t9,/~) for (p,/3) in (2) is proposed as an estimator of ,:.. 2.2 Life test under progressive stress When life test is conducted under progressive stress conditions, maximum likelihood estimators of p and /~ are obtained. An estimator of 2 is then obtained by substituting these maximum likelihood estimators forp and [~in the expression for the posterior mean of 2 given the test data.

2.2.1 Test procedure The m units are put on test and the stress is increased linearly with time from initial stress 0, that is at the rate of pt until all units fail. 2.2.2 Parameter estimation Following Yin and Sheng, 6 the life distribution of units under progressive stress is given as follows:

= 1 - exp{-2[/3/(p

+/3)]~(p/Vu)r(.vP ~)}

(3)

From the conditional distribution of failure times given 2 and the prior distribution of 2, the likelihood function L(p,/3;Yl . . . . . . v,.) is given by

L(p,/3; .vl . . . . . . v,,) = ['~ L(p, [1, )4 )'~ . . . . . . v,.) d2 t M

( p + / 3 ) y ; "t~+~ " = ~ 1 ~ /.[/3/(p+/3)]¢(p/l,o)'P ~ , 3o 1 1 j-:! x exp { - 2 [/3/(p + fl)]~(p/Vu)p ()/,p+t~)] x [h "',.'I(a)]z " "" 1 exp ( - h).) d2

An accelerated life test model with the inverse power law

227

?¢1

= [/3/(p + fl)]"~(p/Vu),.p(p +/3),.

yf+O- lb.F( n + a) j=l m

F(a) b + [B/(p + B)]~(p/Vo) p

(4)

yf÷O- ~ j=!

The

posterior

distribution

of 2 is obtained

as

the

ratio

of

L(p,B,2;y! ..... Ym) to L(p,/3;y 1..... Ym) and turns out to be a gamma distribution with parameters { b + [ / 3 / ( p + f l ) ] ~ ( p / V ~ ) P ~ ) ~ +~-'}

and

(n+a)

j=l

The posterior mean of 2 and is given by

Ep.~[;.;y~ ..... ym] =(n + a)

/f

b+[['t/(p+[~)]~(p/Vu) p

i

2

)~'~-!

j=l

t

(5)

which is the Bayes estimator of 2 under squared error loss when p and fl are known constants. The maximum likelihood estimator (/~,/~) of (p, fl) is obtained by solving the two partial derivatives of the log L(p, fl; Yl . . . . . y,,) with respect to p and/3 simultaneously. E~,~[).; yl ..... y,,] obtained by substituting (/~, fl) for (p,/3) in (5) is proposed as an estimator of 2.

3 A N U M E R I C A L EXAMPLE In this section a numerical example is given and Monte Carlo study is performed to investigate the properties of estimators of p,/3 and 2 under multiple constant stress conditions. We consider Nelson's data 7 as a TABLE 1 Least Square and Proposed Estimates of )., p and fl

Parameters estimated

).

p

[J

Estimates using least square method Proposed estimates G(5000, 1) G(10000,2)

1"414 × 10 -4

16.39

0"8080

1.473 × 10 -4 1.638×10 4

13.30 13.13

0'761 6 0.7561

228

I). S. Bat, S. 14'. ('hung,

numerical example. The data consist of times to breakdown of an insulating fluid subject to various elevated test voltages and are given in the Appendix. Assuming that the parameters c and p of the inverse power law ~./,,r and the shape parameter [~¢are unknown constants, he obtained the least square estimators, c*,p*, fl*.'~ Here we analyze the data assuming that the scale parameter ). of the Weibull distribution at use condition (20kV) has a gamma prior distribution. Table 1 shows various estimates of 2,p and ft. The first row contains the least square estimates p* and fl* o f p and [/, and the estimate c*(20¢'" of 2. The last two rows contain the proposed estimates based on the gamma prior distributions G(5000, 1) and G(10000, 2) for the scale parameter of the Weibull life distribution at use condition. The estimates o f p and [ / a r e computed using the Newton Raphson method.

TABI.E 2 Ratio of the MSE of the Proposed Estimators to the MSE of the Estimators using l,east Square Method (p, [I) parameters

),

p

/I

( 13"0. 0"70)

0"047 0'053

0"347 0.315

0'387 0'340

(I 6'0, 0"80)

0-047 0'052

0-335 0'299

0"387 0-340

{18.0, 0-70)

0.047 0"052

0"331 0.292

0"387 0"340

To see the performance of the proposed estimators over the estimators using least square method, Monte Carlo study is undertaken under the accelerated life test which has the same stress levels and number of units at each level as the life test in the Appendix. Table 2 lists the ratio of the mean squared errors (MSE) of the proposed estimators to the MSE of estimators using least square method. The values of p and fl are selected within the range of values o f p and fl in Table 1. The upper value of each entry of the table is the ratio for the G(5000, 1) prior distribution of). and the lower one is for G(10 000, 2). The table indicates that the proposed estimators perform better than the estimators using the least square method. And the improvement is more pronounced for the case of 2. Numerical studies also reveal that the MSE of the proposed estimator offl is an increasing function of fl only and does not depend on p, and similarly for the MSE of the proposed estimator o f p .

An accelerated life test model with the inverse power law

229

REFERENCES !. Proschan, F. & Singpurwalla, N. D., Accelerated life testing--a pragmatic Bayesian approach. In Optimizing Methods in Statistics, ed. J. S. Rustagi. Academic Press, New York, 1979, pp. 385~,01. 2. Proschan, F. & Singpurwalla, N. D., A new approach to inference from accelerated life tests. I E E E Transactions on Reliability, R-29(2) (1980) 98-102. 3. Shaked, M. & Singpurwalla, N. D., Nonparametric estimation and goodness-offit testing of hypotheses for distributions in accelerated life testing. I E E E Transactions on Reliability, R-31(I) (1982) 69-74. 4. DeGroot, M. H. & Goel, P. K., Bayesian estimation and optimal designs in • partially accelerated life testing. Naval Research Logistics QuarterO', 26(2) (1979) 223--35. 5. Pathak, P. K. & Zimmer, W. J., A Bayesian approach to accelerated life testing. I E E E Proceedings on Reliability and Maintainabilio, (1981), pp. 371-4. 6. Yin, X. & Sheng, B., Some aspects of accelerated life testing by progressive stress. I E E E Transactions on Reliabilio; R-36(! ) (I 987) 150- 5. 7. Nelson, W., Graphical analysis of accelerated life test data with the inverse power law model. I E E E Transactions on Reliabilio', R-21(1 ) (1975) 2-11. 8. Nelson, W., Analysis of accelerated life test data--least squares methods for the inverse power law model. I E E E Transactions on Reliability, R-24(2) (1975) 103 -7.

230

D..";. Bai, S. W. Chune

A P P E N D I X : N E L S O N ' S DATA Nelson's data 7 is reproduced for reference. Table A I shows the stress levels, the failure times and the number of items tested at each stress level. TABLE A! Brcakdov,n Data of Insulating Fluid (Nelson') V,

26 k V

28 k |"

30 k J

32 k l"

t~j

579 I 57952 2 323-70

68'85 108"29 110-29 426.07 I 06760

7.74 17-05 2046 21'02 22.66 43.40 47.30 139.07 144.12 175.88 194.90

0.27 0.40 0.69 079 2'75 3.91 9"88 1395 1593 2780 53.24 82.85 89.29 100.58 21510

n,

3

5

11

15

34 k l'"

36 k V

38 k ["

0 19 0"78 0"96 1'31 2-78 3 16 4-15 4.67 4"85 6'50 7-35 801 8-27 12'06 3175 32'52 3391 36"71 7289

0"35 0.78 0.96 0-99 1'69 1.97 2.07 258 271 2-90 3.67 3.99 535 13.77 2550

0"09 0"39 047 073 0.74 I 13 1'40 238

It;

15

8